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Any-trace

Any-trace-equiv

Basic equivalence relation for any-trace structures.

Definitions and Theorems

Function: any-trace-equiv$inline

(defun any-trace-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (any-trace-p acl2::x)
                              (any-trace-p acl2::y))))
  (equal (any-trace-fix acl2::x)
         (any-trace-fix acl2::y)))

Theorem: any-trace-equiv-is-an-equivalence

(defthm any-trace-equiv-is-an-equivalence
  (and (booleanp (any-trace-equiv x y))
       (any-trace-equiv x x)
       (implies (any-trace-equiv x y)
                (any-trace-equiv y x))
       (implies (and (any-trace-equiv x y)
                     (any-trace-equiv y z))
                (any-trace-equiv x z)))
  :rule-classes (:equivalence))

Theorem: any-trace-equiv-implies-equal-any-trace-fix-1

(defthm any-trace-equiv-implies-equal-any-trace-fix-1
  (implies (any-trace-equiv acl2::x x-equiv)
           (equal (any-trace-fix acl2::x)
                  (any-trace-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: any-trace-fix-under-any-trace-equiv

(defthm any-trace-fix-under-any-trace-equiv
  (any-trace-equiv (any-trace-fix acl2::x)
                   acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-any-trace-fix-1-forward-to-any-trace-equiv

(defthm equal-of-any-trace-fix-1-forward-to-any-trace-equiv
  (implies (equal (any-trace-fix acl2::x) acl2::y)
           (any-trace-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-any-trace-fix-2-forward-to-any-trace-equiv

(defthm equal-of-any-trace-fix-2-forward-to-any-trace-equiv
  (implies (equal acl2::x (any-trace-fix acl2::y))
           (any-trace-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: any-trace-equiv-of-any-trace-fix-1-forward

(defthm any-trace-equiv-of-any-trace-fix-1-forward
  (implies (any-trace-equiv (any-trace-fix acl2::x)
                            acl2::y)
           (any-trace-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: any-trace-equiv-of-any-trace-fix-2-forward

(defthm any-trace-equiv-of-any-trace-fix-2-forward
  (implies (any-trace-equiv acl2::x (any-trace-fix acl2::y))
           (any-trace-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)